Inversion and subspaces of a finite field

Abstract

Let $A$ and $B$ two $F_q$-subspaces of a finite field, of the same size, and let $A^{-1}$ denote the set of inverses of the nonzero elements of $A$. Mattarei proved that $A^{-1}$ can only be contained in $A$ if either $A$ is a subfield, or $A$ is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if $A^{-1}⊈B$, then the bound $|A^{-1}\cap B|\le 2|B|/q-2$ holds. He also gave examples showing that his bound is sharp for $|B|\le q^3$. Our main result is a proof of the stronger bound $|A^{-1}\cap B|\le |B|/q\cdot (1+O_d(q^{-1/2}))$, for $|B|=q^d$ with $d>3$. We also classify all examples with $|B|\le q^3$ which attain equality in Csajbók's bound.

20 pages. Changes from previous version: added comment on similar bound for A and B affine subspaces, in Section 2; added Remark 11, and Reference, on alternate approach to when Csajbok's bound is attained for |B|=q^3, via facts from finite geometries. Version submitted for publication

References in corpus (2)

• The Kakeya set and maximal conjectures for algebraic varieties over finite fields
• A property of the inverse of a subspace of a finite field

Cited by in corpus (1)

• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II